# Properties

 Label 2475.2.c.n Level $2475$ Weight $2$ Character orbit 2475.c Analytic conductor $19.763$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(199,2475)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2475.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2475 = 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2475.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$19.7629745003$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 165) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} - q^{4} + \beta_1 q^{7} + \beta_{2} q^{8}+O(q^{10})$$ q + b2 * q^2 - q^4 + b1 * q^7 + b2 * q^8 $$q + \beta_{2} q^{2} - q^{4} + \beta_1 q^{7} + \beta_{2} q^{8} + q^{11} + ( - 2 \beta_{2} - \beta_1) q^{13} - \beta_{3} q^{14} - 5 q^{16} + ( - \beta_{3} - 2) q^{19} + \beta_{2} q^{22} + 4 \beta_{2} q^{23} + (\beta_{3} + 6) q^{26} - \beta_1 q^{28} - \beta_{3} q^{29} + ( - 2 \beta_{3} - 4) q^{31} - 3 \beta_{2} q^{32} + ( - 4 \beta_{2} + \beta_1) q^{37} + ( - 2 \beta_{2} - 3 \beta_1) q^{38} - \beta_{3} q^{41} + (4 \beta_{2} - \beta_1) q^{43} - q^{44} - 12 q^{46} + 4 \beta_{2} q^{47} + 3 q^{49} + (2 \beta_{2} + \beta_1) q^{52} + (4 \beta_{2} - 3 \beta_1) q^{53} - \beta_{3} q^{56} - 3 \beta_1 q^{58} - 2 \beta_{3} q^{59} + 2 q^{61} + ( - 4 \beta_{2} - 6 \beta_1) q^{62} - q^{64} + 4 \beta_1 q^{67} - 4 \beta_{3} q^{71} + (6 \beta_{2} - \beta_1) q^{73} + ( - \beta_{3} + 12) q^{74} + (\beta_{3} + 2) q^{76} + \beta_1 q^{77} + ( - \beta_{3} + 10) q^{79} - 3 \beta_1 q^{82} + ( - 2 \beta_{2} + 6 \beta_1) q^{83} + (\beta_{3} - 12) q^{86} + \beta_{2} q^{88} + (2 \beta_{3} - 6) q^{89} + (2 \beta_{3} + 4) q^{91} - 4 \beta_{2} q^{92} - 12 q^{94} - 5 \beta_1 q^{97} + 3 \beta_{2} q^{98}+O(q^{100})$$ q + b2 * q^2 - q^4 + b1 * q^7 + b2 * q^8 + q^11 + (-2*b2 - b1) * q^13 - b3 * q^14 - 5 * q^16 + (-b3 - 2) * q^19 + b2 * q^22 + 4*b2 * q^23 + (b3 + 6) * q^26 - b1 * q^28 - b3 * q^29 + (-2*b3 - 4) * q^31 - 3*b2 * q^32 + (-4*b2 + b1) * q^37 + (-2*b2 - 3*b1) * q^38 - b3 * q^41 + (4*b2 - b1) * q^43 - q^44 - 12 * q^46 + 4*b2 * q^47 + 3 * q^49 + (2*b2 + b1) * q^52 + (4*b2 - 3*b1) * q^53 - b3 * q^56 - 3*b1 * q^58 - 2*b3 * q^59 + 2 * q^61 + (-4*b2 - 6*b1) * q^62 - q^64 + 4*b1 * q^67 - 4*b3 * q^71 + (6*b2 - b1) * q^73 + (-b3 + 12) * q^74 + (b3 + 2) * q^76 + b1 * q^77 + (-b3 + 10) * q^79 - 3*b1 * q^82 + (-2*b2 + 6*b1) * q^83 + (b3 - 12) * q^86 + b2 * q^88 + (2*b3 - 6) * q^89 + (2*b3 + 4) * q^91 - 4*b2 * q^92 - 12 * q^94 - 5*b1 * q^97 + 3*b2 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4}+O(q^{10})$$ 4 * q - 4 * q^4 $$4 q - 4 q^{4} + 4 q^{11} - 20 q^{16} - 8 q^{19} + 24 q^{26} - 16 q^{31} - 4 q^{44} - 48 q^{46} + 12 q^{49} + 8 q^{61} - 4 q^{64} + 48 q^{74} + 8 q^{76} + 40 q^{79} - 48 q^{86} - 24 q^{89} + 16 q^{91} - 48 q^{94}+O(q^{100})$$ 4 * q - 4 * q^4 + 4 * q^11 - 20 * q^16 - 8 * q^19 + 24 * q^26 - 16 * q^31 - 4 * q^44 - 48 * q^46 + 12 * q^49 + 8 * q^61 - 4 * q^64 + 48 * q^74 + 8 * q^76 + 40 * q^79 - 48 * q^86 - 24 * q^89 + 16 * q^91 - 48 * q^94

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$2\zeta_{12}^{3}$$ 2*v^3 $$\beta_{2}$$ $$=$$ $$2\zeta_{12}^{2} - 1$$ 2*v^2 - 1 $$\beta_{3}$$ $$=$$ $$-2\zeta_{12}^{3} + 4\zeta_{12}$$ -2*v^3 + 4*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 4$$ (b3 + b1) / 4 $$\zeta_{12}^{2}$$ $$=$$ $$( \beta_{2} + 1 ) / 2$$ (b2 + 1) / 2 $$\zeta_{12}^{3}$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times$$.

 $$n$$ $$551$$ $$2026$$ $$2377$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 0.866025 − 0.500000i −0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
1.73205i 0 −1.00000 0 0 2.00000i 1.73205i 0 0
199.2 1.73205i 0 −1.00000 0 0 2.00000i 1.73205i 0 0
199.3 1.73205i 0 −1.00000 0 0 2.00000i 1.73205i 0 0
199.4 1.73205i 0 −1.00000 0 0 2.00000i 1.73205i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2475.2.c.n 4
3.b odd 2 1 825.2.c.c 4
5.b even 2 1 inner 2475.2.c.n 4
5.c odd 4 1 495.2.a.c 2
5.c odd 4 1 2475.2.a.r 2
15.d odd 2 1 825.2.c.c 4
15.e even 4 1 165.2.a.b 2
15.e even 4 1 825.2.a.e 2
20.e even 4 1 7920.2.a.bz 2
55.e even 4 1 5445.2.a.s 2
60.l odd 4 1 2640.2.a.x 2
105.k odd 4 1 8085.2.a.bd 2
165.l odd 4 1 1815.2.a.i 2
165.l odd 4 1 9075.2.a.bh 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.a.b 2 15.e even 4 1
495.2.a.c 2 5.c odd 4 1
825.2.a.e 2 15.e even 4 1
825.2.c.c 4 3.b odd 2 1
825.2.c.c 4 15.d odd 2 1
1815.2.a.i 2 165.l odd 4 1
2475.2.a.r 2 5.c odd 4 1
2475.2.c.n 4 1.a even 1 1 trivial
2475.2.c.n 4 5.b even 2 1 inner
2640.2.a.x 2 60.l odd 4 1
5445.2.a.s 2 55.e even 4 1
7920.2.a.bz 2 20.e even 4 1
8085.2.a.bd 2 105.k odd 4 1
9075.2.a.bh 2 165.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2475, [\chi])$$:

 $$T_{2}^{2} + 3$$ T2^2 + 3 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{29}^{2} - 12$$ T29^2 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 3)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 4)^{2}$$
$11$ $$(T - 1)^{4}$$
$13$ $$T^{4} + 32T^{2} + 64$$
$17$ $$T^{4}$$
$19$ $$(T^{2} + 4 T - 8)^{2}$$
$23$ $$(T^{2} + 48)^{2}$$
$29$ $$(T^{2} - 12)^{2}$$
$31$ $$(T^{2} + 8 T - 32)^{2}$$
$37$ $$T^{4} + 104T^{2} + 1936$$
$41$ $$(T^{2} - 12)^{2}$$
$43$ $$T^{4} + 104T^{2} + 1936$$
$47$ $$(T^{2} + 48)^{2}$$
$53$ $$T^{4} + 168T^{2} + 144$$
$59$ $$(T^{2} - 48)^{2}$$
$61$ $$(T - 2)^{4}$$
$67$ $$(T^{2} + 64)^{2}$$
$71$ $$(T^{2} - 192)^{2}$$
$73$ $$T^{4} + 224 T^{2} + 10816$$
$79$ $$(T^{2} - 20 T + 88)^{2}$$
$83$ $$T^{4} + 312 T^{2} + 17424$$
$89$ $$(T^{2} + 12 T - 12)^{2}$$
$97$ $$(T^{2} + 100)^{2}$$